Cross-product term

Sometimes potential energy surfaces are plotted with skewed axes that is, the Tab 2nd tbc axes meet at an angle less than 90°. This is done so that the relative kinetic energy of the three-body system can be represented by the motion of a single point over the surface. In order to achieve this condition it is necessaiy that the cross-product terms in the kinetic energy drop out. The calculations have been described - Because our use of potential energy surfaces is qualitative,... 

Computation of the cross-product term in the pooled variance-covariance matrix for the data of Table 33. 

Multiplication of (1) and (2) gives rise to a cross-product term of the form ... 

The technique allows immediate interpretation of the regression equation by including the linear and interaction (cross-product) terms in the constant term (To or stationary point), thus simplifying the subsequent evaluation of the canonical form of the regression equation. The first report of canonical analysis in the statistical literature was by Box and Wilson [37] for determining optimal conditions in chemical reactions. Canonical analysis, or canonical reduction, was described as an efficient method to explore an empirical response surface to suggest areas for further experimentation. In canonical analysis or canonical reduction, second-order regression equations... 

Equation 41-A3 can be checked by expanding the last term, collecting terms and verifying that all the terms of equation 41-A2 are regenerated. The third term in equation 41-A3 is a quantity called the covariance between A and B. The covariance is a quantity related to the correlation coefficient. Since the differences from the mean are randomly positive and negative, the product of the two differences from their respective means is also randomly positive and negative, and tend to cancel when summed. Therefore, for independent random variables the covariance is zero, since the correlation coefficient is zero for uncorrelated variables. In fact, the mathematical definition of uncorrelated is that this sum-of-cross-products term is zero. Therefore, since A and B are random, uncorrelated variables ... 

The first two terms on the RHS of equation 70-20 are the variances of X and Y. The third term, the numerator of which is known as the cross-product term, is called the covariance between X and Y. We also note (almost parenthetically) here that multiplying both sides of equation 70-20 by (re - 1) gives the corresponding sums of squares, hence equation 70-20 essentially demonstrates the partitioning of sums of squares for the multivariate case. 

And note a fine point we have deliberately ignored until now that in equation 70-20 the (Statistical) cross-product term was multiplied by two. This translates into the two appearances of that term in the (chemometrics) cross-product matrix. 

The cross-product term P A)P B) compensates for counting the overlapping cases twice. Consider the example of tossing a single die and determining the probability that the number of points is even or divisible by 3. In this case... 

The other procedure is to use tl e minimal cut sets. This procedure approaches the exact result only if the probabilities of all the events are small. In general, this result provides a number that is larger than the actual probability. This approach assumes that the probability cross-product terms shown in Equation 11-10 are negligible. 

This compares to the exact result of 0.0702 obtained using the actual fault tree. The cut sets are related to each other by the OR function. For Example 11-6 all the cut set probabilities were added. This is an approximate result, as shown by Equation 11-10, because the cross-product terms were neglected. For small probabilities the cross-product terms are negligible and the addition will approach the true result. 

For several covariates we simply introduce a cross-product term for each covariate with corresponding coefficients d, 2 and dj. The presence of treatment-by-covariate interactions can then be investigated through these coefficients. 

We can also investigate the presence of treatment-by-covariate interactions by including cross-product terms ... 

Consider, now, a second example where the response from the experiment can be adequately modeled by a model that contains linear terms in the design variables, x, and the environmental variables, z, and also cross-product terms xz. Therefore, if there are n design variables, x, x, .. ., x, and m environmental variables, z, z, .. ., z, then the response, y, can be represented by... 

Equation 1 predicts the vaiues of the response by considering only the purely linear effects of the variables. Equation 2 employs linear, quadratic, and cross-product terms to produce a better prediction of the response (narrower limits of variation). 

An even more complex model is possible for this situation. It includes all possible linear, quadratic, and linear-quadratic cross-product terms. 

Artificial neural networks learn the correct constituent classification model through iterative trial-and-error calculations to determine which frequencies in the data show the best ability to classify the pixels according to the constituent type. They can explore nonlinear as well as linear relationships among the frequencies by using, for example, squared and cross-product terms of the frequency data. However, because of the extreme nonlinearities that may be found in a neural net model, their results are often uninterpretable in a classical sense, even though they may predict quite well. 

The vector product or "cross" product (term coined by Gibbs53) is defined only in three-dimensional space The vector product, or cross product, of vectors a and b is a vector v, whose magnitude is a b sin y, where y is the angle between a and b, and whose direction is perpendicular to both a and b, and whose orientation is such that a, b, and v form a right-handed system ... 

The contribution of the 2T2g(t2g) level to the Jahn-Teller activity is therefore nil and any cross product terms vanish due to spin orthogonality so that only the 4T2g(t2geJ) state need be considered. Thus, by perturbation theory, one may to a good approximation write the rm component (m= x, A, fi, v) of the r8 d3 O ground state as... 

In the second-order interaction model cross-product terms have also been included. A significant estimated cross-product coefficient, bVj, shows that the influence of variable X is dependent on the settings of variable xp i.e. there is an interaction effect between these variables. Geometrically this corresponds to a twist of the response surface over the xi x3 plane. 

The rotation removes the cross-product terms from the original response surface model. This form of the canonical model is very useful for exploring ridge systems. Such systems occur when some eigenvalue is small due to a feeble curvature of the surface in the corresponding Z direction. The variation in this direction is therefore largely described by the corresponding linear coefficient 9,. 

To rotate the axes through an angle a, substitute for x the quantity X cos a -sin a and for y the quantity x sin a -H cos a. A rotation of the axes through a = cot (A — C)/B will eliminate the cross-product term in the general second-degree equation. 

If the response surface looks like the one given in Fig. 3.4., a linear model will not yield a good description. The response surface is a twisted plane. The slope of the surface along the Xj axis will depend on the value of variable X2- This means that the influence of variable Xj will depend on the settings of variable X2, i.e. there is an interaction effect between Xj and X2. In such cases, the model will be improved if a cross-product term, 653X5X2 is included ... 

Pig. 3.4 A second-order interaction model can describe interactions between tbe experimental variables. The surfece is a twisted plane. The twist is described by the cross-product term. 

This will provide us with a first check of which variables are influencing the response. It is then always possible to run a complementary set of new experiments to augment the model by cross-product terms to allow for an analysis of interaction effects, i.e. 

Tbe vector Zj is the vector of all variables in the model. This vector is obained from Xj by adding the elements which correspond to the constant term, cross-product terms, square terms. .. etc. 

The objectives of a screening experiment is to proceed from a stage where very little is known with certainty about the roles played by the experimental variables, up to a stage where the relative importance of the variables can be assessed. In synthetic chemistry this entails the identification of those variables which have a significant influence on, for example, yield and selectivity. To achieve this, an approximate response function is established. This model contains linear terms of the variables of interest, and possibly also cross-product terms to describe interaction effects. 

Cross product terms, like e.g. MRj MR2 MR3 [456], should be avoided because they are highly interrelated with squared terms (e.g. MR, MR2 MRj us. a combination of ZMR, MRj, MR2, and MR3 n = 71 r = 0.993) [175]. Curious nonlinear models, like hyperbolic regressions [422, 457], sinus [457], or tangens terms [458] shall be mentioned here without giving advice to use them these approaches can only be characterized by a phrase, created by Hansch, as "statistical unicorns, beasts that exist on paper but not in reality [307]. 

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